New Civilization News: Pascal's Triangle, Self-similarity and Phi    
 Pascal's Triangle, Self-similarity and Phi4 comments
23 Apr 2008 @ 09:25, by John Grieve

Pascal’s Triangle, Self-Similarity and Phi

In maths the simple operation of adding two consecutive elements in a sequence and then iterating, which process is well-known to us in the Fibonacci sequence, leads to many of the more remarkable properties we come across in nature and mathematics. The Fibonacci sequence, as I pointed out in my last article, is based on self-similarity and exhibits the mystical number and proportion called the “Sacred Ratio”, approximately 1.618… which is an irrational number.

Now something very similar occurs in that very famous table of numbers, known to the ancient Chinese, but known to us as Pascal’s Triangle. This is a symmetrical table, with ones at the apex and at each edge, with the intervening numbers created by adding together the two numbers directly above and to either side. Pascal’s triangle has a multitude, maybe indeed an infinite number of remarkable properties. Every interesting thing in mathematics more or less, can be found in different ways in this pyramid. Fibonacci itself, can be found in sequence if you add the short diagonals. This of course yields Phi, the Golden Proportion. However, this is a bit misleading, because Phi is conspicuously absent from the other patterns you will find in this triangle. This is because if you divide any two of the numbers in the table they will be a rational fraction, not irrational. Adding different numbers together, as in the Fibonacci example, is the only way to get a sequence which gives Phi. The whole structure is based on the iterative technique mentioned above, and I suspect that this technique is a cornerstone of self-similarity, though I can’t demonstrate it as convincingly here as I did in my previous article on Fibonacci.

I believe, since Nature produces Phi all over the place, and Fibonacci sequences in the number of petals of flowers and the spirals of shells, that at an early stage in the evolution of life, in plant RNA and animal DNA, the simple iterative technique I refer to, was encoded and passed down to following generations. Thus we find Phi everywhere in Nature. Why it leads to very remarkable properties in mathematics is another issue and one I will address in a later article.

[< Back] [New Civilization News]



27 May 2008 @ 20:16 by mortimer : Hi John, nice Fibonacci Rhythms..
I thought this would be good place to post the programmers Fibonacci algorithm,


Mathematician Leonardo Fibonacci posed the following problem in his treatise Liber Abaci: "How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?"

The answer is given below by algorithms in various programming languages...

also, {| Fractal Tune Smithy} is my favorite music software for exploring the Fibonacci rhythms
it even has a mouse theremin, cool ..

18 Mar 2015 @ 10:24 by foodiefruitgarden @ : Person, I truly adored scanning this art
Person, I truly adored scanning this article. You have convinced my family to join towards your site, yet everywhere can I find the RSS feed?  

2 Jan 2016 @ 14:09 by Jugar Juegos Gratis @ : juegos online
Diviértete con los mejores juegos de Disney Channel, Disney XD, películas Disney y mucho más!  

29 Apr 2016 @ 05:24 by Bandar Togel @ : brilliant! I would like to share this ar
Togel Online Singapore
Togel Online Hongkong
Bandar Togel Singapore
Bandar Togel
Togel Online Terpercaya
Bandar Togel Online Terpercaya
Togel Online
Agen Togel Online Terpercaya
Agen Togel Online  

Your Name:
Your URL: (or email)
For verification, please type the word you see on the left:

Other entries in
28 Jun 2010 @ 00:03: Pump up the synchronicity
27 Jun 2010 @ 02:28: Be afraid, be very afraid
23 Nov 2008 @ 22:54: Endspiel - der Monopol und Deine Aufhebung
20 Oct 2008 @ 18:48: For discussion: Revised Call for Papers & logo
7 May 2008 @ 09:27: What is a system and why should we care to know?
4 May 2008 @ 01:08: System Oriented Modelling Paradigm - Brainstorming notes 03
25 Apr 2008 @ 11:32: System Oriented Modelling Paradigm
13 Apr 2008 @ 09:47: Phi is the constant of Self-Similarity
1 Dec 2007 @ 16:36: considering options
12 Jul 2007 @ 22:53: Emergence and democracy

[< Back] [New Civilization News] [PermaLink]?